The number of students in Teresa's graduating class is more than 50 and fewer than 100 and is 1 less than a multiple of 3, 2 less than a multiple of 4, and 3 less than a multiple of 5. How many students are in Teresa's graduating class?
Solution: Let the number of students in Teresa's class be $a$. Then \begin{align*}
a\equiv -1\equiv 2\pmod 3,\\
a\equiv -2\equiv 2\pmod 4,\\
a\equiv -3\equiv 2\pmod 5.
\end{align*} Since $\gcd(3,4)=\gcd(4,5)=\gcd(3,5)=1$, we have $$a\equiv 2\pmod{3\cdot 4\cdot 5},$$ that is, $a\equiv 2\pmod{60}$. Since $a$ is then of the form $a=2+60n$, the only such number in the range $50<a<100$ is $\boxed{62}$.